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Particle conserving approach to nuclear pairing

Alexander Volya, Vladimir Zelevinsky

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We discuss a method for treating the general pairing problem in a nite Fermisystem without violating the particle number conservation.The operator equations of motion are brought to the form of the recurrence relations with respect to the particle number.Examples of applications are shown.The results reproduce the exactly solvable models and consistently display an advantage compared to the conventional BCS solution.Extensions of the method are discussed.

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We discuss a method for treating the general pairing problem in a nite Fermisystem without violating the particle number conservation.The operator equations of motion are brought to the form of the recurrence relations with respect to the particle number.Examples of applications are shown.The results reproduce the exactly solvable models and consistently display an advantage compared to the conventional BCS solution.Extensions of the method are discussed.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.
An equations-of-motion method is proposed for treating neutron-proton pairing correlations conserving both nucleon number NNN and isospin TTT. This method makes it possible to find approximate solutions to the charge-independent pairing Hamiltonian by a step-by-step procedure. It is shown that for the degenerate model the exact results are obtained.NUCLEAR STRUCTURE Charge-independent treatment of pairing correlations.
The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.
A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.
A practical number-conserving method of treating pairing correlations is proposed. The theory is formulated in a form having clear correspondence with quasiparticle theory and, moreover, it works well even in the weak pairing limit. Various matrix elements are approximately expressed in terms of the occupation probabilities containing the blocking effects within the number-conserving framework. The exact occupation probabilities can be used in the case of a constant pairing force. The present method is able to take the place of previous fixed-number BCS methods and is suitable for treating residual correlations further. Application to the cranking model is discussed.